According to these notes by Yilong Zhang the function $x\mapsto e^x$ is not definable in any interval in the $o$-minimal structure $\Bbb R_{\mathrm{an}}$, where the latter is defined to be the structure on $\Bbb R$ generated by $$\mathcal F_{\mathrm{an}}=\{f\mid f=g_{|_{[-1,1]^n}},\text{ $g$ is a real analytic function defined in a nbhd of $[-1,1]^n$}\}.$$
Based on this definition it seems clear to me that $x\mapsto e^x$ is definable in this structure in the interval $[-1,1]$. I guess that the correct claim should be that $x\mapsto e^x$ is not definable in $\Bbb R_{\mathrm{an}}$ as a function defined over the whole of $\Bbb R$.
Is my understanding correct? And if so how does one see that $x\mapsto e^x$ is not definable in $\Bbb R_{\mathrm{an}}$ as function $\Bbb R\to \Bbb R$?
As the OP comments below, the relevant line is just before example $2$: "For example, $x\mapsto\sqrt{1-x^2}$ for $\vert x\vert\le 1$ is $\mathbb{R}_{an}$-definable, but $x\mapsto e^x$ is not (in any interval)."
This is of course a typo, since e.g. $x\mapsto e^x$ on the interval $[-1,1]$ is literally built into the structure $\mathbb{R}_{an}$ at the signature level. However, note that this sentence occurs in the context of discussing $\mathbb{R}_{sa}$, and before $\mathbb{R}_{an}$ has been defined in the first place. So I think the typo is actually "$\mathbb{R}_{an}$" for "$\mathbb{R}_{sa}$," not an erroneous mention of intervals.
Supporting this, note that later on Zhang says (top of page $3$) "The function $x\mapsto \arctan(x)$, $x\in\mathbb{R}$ is $\mathbb{R}_{an}$-definable, but $x\mapsto e^x, x\in\mathbb{R}$ is not."
Meanwhile, if memory serves the proof of the undefinability of (the whole of) $x\mapsto e^x$ in $\mathbb{R}_{an}$ is extremely difficult.